Directed Q-Analysis and Directed Higher-Order Connectivity on Digraphs: A Quantitative Approach

TL;DR

This paper introduces a mathematical framework for analyzing higher-order directed relationships in networks, extending traditional graph analysis to include complex multi-node interactions.

Contribution

It develops a formalism based on directed clique complexes to quantify and compare higher-order structures in directed networks, advancing directed Q-analysis.

Findings

Defines directed clique complexes for directed graphs.

Establishes a method to quantify higher-order connectivity.

Enables comparison of simplicial structures in directed networks.

Abstract

Traditional graph analysis focuses on nodes and edges, that is, pairwise relationships. Yet many real-world networks, including biological, social, and communication networks, involve higher-order relationships in which multiple nodes interact simultaneously. This has led many to develop network topology analysis methods based on higher-order structures and higher-order connectivity, seeking to reveal complex interactions beyond node pairs. Many of the latter address only undirected networks. To overcome this, we lay out a mathematical formalism resting on directed clique complexes constructed from directed graphs (their "higher-order structures" or "simplicial structures''), stressing the interrelations between directed cliques (their "directed higher-order connectivities''), leading towards a more complete directed Q-analysis that allows quantifying, characterizing, and comparing similarities involving simplicial structures.